Stationary types with non-stationary extensions

Question

Let $T$ be a complete stable theory (say in a countable language $L$). Let $\mathbb{M}$ be a monster model of $T$ and let $A\subseteq{B}$ be a small subsets. Let $p\in S(A)$ be stationary. Let $q\in S(B)$ be be such that $q$ extends $p$.

If $q$ is an non-forking extension of $p$ it follows that $q$ is also stationary. This seems not to generalize if $q$ is a forking extension.

Given the above setup, what is an example of a theory $T$ such that $q$ is a forking extension of $p$ and $q$ is not stationary? I tried the usual "equivalence" class examples, but the ones I tried end up making $q$ stationary because of nature of the example itself. Examples like $ACF_0$ don't seem to work either.

Answer 1

Levon gave you an example in $ACF$. I'll give you an example in equivalence relations.

Let $E$ be an equivalence relation with infinitely many classes, each of size $3$. Consider the type $p(x)$ over $A$ which says that $x$ is not $E$-related to any element of $A$. This type is stationary. Now let $B = A\cup \{b\}$, where $b$ is an element of a class not represented in $A$. $p$ has a forking extension $q$ over $B$ which contains the formulas $xEb$ and $x\neq b$. This type is not stationary, because it is algebraic with two realizations (just like Levon's example).

Answer 2

Consider the following example in $ACF_p$ for $p \neq 2$. Let $A$ be a field and let $a$ be a transcendental element over $A$. Then $tp(a/A)$ is stationary. But $tp(a/Aa^2)$ isn't: it is isolated by the formula $x^2 = a^2$ which has Morley degree $2$. If you like to think in terms of extensions $tp(a/Aa^2a)$ and $tp(-a/Aa^2a)$ are two nonforking extensions of $tp(a/Aa^2)$.